Theorem REF  can be regarded as the generalization of subsection 3.18 of [1]} and Theorem 21 of [2]}. Actually, the range of the order can be wider. If \(\nabla ^{n,w(k)} x(k)\)  is nonnegative, then for any \(\alpha \in (n-1,n)\) , \(n\in \mathbb {Z}_+\) , \(w:\mathbb {N}_a\rightarrow \mathbb {R}\backslash \lbrace 0\rbrace \)  with \(\frac{w(a)}{w(k)}>0\) , \(k\in \mathbb {N}_{a+1}\) , \(a\in \mathbb {R}\) , one has \({}_a^{\rm C}\nabla _k^{\alpha ,w(k)}x(k)\)  is nonnegative. It will be more interesting to construct the sufficient condition for \(\nabla ^{1, w(k)} x(k) \ge 0\)  with the assumption \({ }_{a}^{\mathrm {C}} \nabla _{k}^{\alpha , w(k)} x(k) \ge 0\)  or \({ }_{a}^{\mathrm {R}} \nabla _{k}^{\alpha , w(k)} x(k) \ge 0\)
